113 lines
3.2 KiB
Python
113 lines
3.2 KiB
Python
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import math
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import numpy as np
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import matplotlib.pyplot as plt
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import sounddevice as sd
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sample_rate = 44100
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seconds = 5
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N = 100 # number of string segments
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I = int(sample_rate * seconds) # number of samples to simulate
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# string parameters
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rho = 8000 # density, steel, kg/m^3
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radius = 0.001 # meters
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S = math.pi*radius**2 # string cross sectional area, assuming circular
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mu = S*rho # linear mass density
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T = 1200 # string tension, N
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c = math.sqrt(T/mu) # transverse wave velocity
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kappa = 0.001 # stiffness coefficient
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sigma = 0.5 # damping coefficient
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L = 0.5 # length of string
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strike_position = 0.2 # x of impulse location
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impulse_width = 0.02 # x of impulse width
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impulse_velocity = 1000.0 # x/t of impulse magnitude
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sample_position = 0.1 # percentage along L of sampling for audio
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dx = L / N # delta x
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dt = 1/sample_rate
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#dt = 0.2 * dx / c
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if(dx**2 < (c*dt)**2 + 4*(kappa*dt/(dx**2))**2):
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print("warning: possibly unstable due to not enough segments, increase N")
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# derived constants
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r1 = c * dt/dx
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r2 = (c * dt/dx) ** 2
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s1 = kappa * dt/dx**2
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s2 = (kappa * dt/dx**2) ** 2
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# string grid
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x = np.linspace(0, L, N + 1) # linspace my beloved
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# state vectors
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y_last = np.zeros(N + 1)
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y_current = np.zeros(N + 1)
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y_next = np.zeros(N + 1)
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# output
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y_sample = np.zeros(I)
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# vector of initial velocity across the string as a result of the impulse
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v0 = impulse_velocity * np.exp(-((x - strike_position) ** 2)/(2 * impulse_width ** 2))
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# initial conditions:
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# y(x, 0) = 0 and dy/dt(x, 0) = v0(x)
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# boundary conditions: y(0, t) = y(L, t) = 0 as well as all dy/dts thereafter
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def show_plot():
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plt.plot(np.arange(0, N+1, 1), y_current)
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plt.grid()
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plt.show()
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# first iteration
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for n in range(2, N-2):
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y_current[n] = dt * v0[n] + 0.5 * r1**2 * (y_last[n-1] - 2*y_last[n] + y_last[n+1])
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# y_xx = y_last[n+1] - 2*y_last[n] + y_last[n-1]
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# y_xxxx = y_last[n-2] - 4*y_last[n-1] + 6*y_last[n] - 4*y_last[n+1] + y_last[n+2]
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# term1 = (2 - 2*sigma*dt) * y_current[n]
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# term2 = (2*sigma*dt - 1) * y_last[n]
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# term3 = r2 * y_xx - s2 * y_xxxx
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# y_current[n] = dt * v0[n] + 0.5 * r1**2 *(term1 + term2 + term3)
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n_sample = int(sample_position * L * N)
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y_sample[0] = y_last[n_sample]
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y_sample[1] = y_current[n_sample]
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# rest of the iterations
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for i in range(2, I):
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for n in range(2, N-2):
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# simple wave equation
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# y_next[n] = 2*y_current[n] - y_last[n] + r**2 * (y_current[n-1] - 2*y_current[n] + y_current[n+1])
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# stiff wave equation
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y_xx = y_current[n+1] - 2*y_current[n] + y_current[n-1]
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y_xxxx = y_current[n-2] - 4*y_current[n-1] + 6*y_current[n] - 4*y_current[n+1] + y_current[n+2]
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term1 = (2 - 2*sigma*dt) * y_current[n]
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term2 = (2*sigma*dt - 1) * y_last[n]
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term3 = r2 * y_xx - s2 * y_xxxx
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y_next[n] = term1 + term2 + term3
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y_next[0] = 0
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y_next[1] = 0
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y_next[N-1] = 0
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y_next[N-2] = 0
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# y_sample[i] = math.tanh(y_next[n_sample])
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y_sample[i] = y_next[n_sample]
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y_last = y_current.copy()
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y_current = y_next.copy()
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if(i % 1000 == 0):
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print(i/I * 100, "% complete")
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plt.plot(np.arange(0, I, 1), y_sample)
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plt.grid()
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plt.show()
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sd.play(y_sample, sample_rate)
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sd.wait()
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